For a political philosopher to be in love with anyone is a sad thing.

Wonder what the other end of the conversation was like.

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For those who were wondering why there hadn’t been any new posts in a while: this beauty from the shores of Lake Como, that’s why.

]]>- A formula is satisfied by an assignment in if and only if the formula is satisfied, in , by every assignment that that differs from at most in the value .

The detour through satisfaction, while technically interesting, it is just that — a detour. It can also sometimes get pedagogically problematic, as students usually grasp the importance of truth while not always appreciating the necessity of satisfaction.

There is an alternative approach, which does away with the notion of satisfaction, but uses a variant notion of a model. According to this approach, we only allow models that are “rich” in that every object is the denotation of a constant in the language. The truth clause for the universal quantifier is then as follows:

- A sentence is true in if and only if is true in for each constant .

(I believe this definition is fairly standard in some textbooks, but I can’t track down a definition now, any references appreciated). While this definition sidesteps the notion of satisfaction, there are some drawbacks:

- The definition only applies to “rich” models, while models in which some elements are not denoted by constants are also very natural. Moreover, we are forced to consider truth of sentence in a model having a different signature than itself.
- The language changes with the model: for each model , the definition applies only to the expanded language introducing new constants for each member of .
- The definition forces us to consider uncountable languages, in order to assess truth in uncountable models.
- The expanded signature affects the available automorphism of a model. For example, the model has infinitely many automorphisms in the signature with the relation only, but only one if we add the constant .

So here is a way of combining these two approaches in such a way that:

- There is no detour through satisfaction.
- The language is countable and independent of the model in which a sentence is evaluated for truth (although the language will be expanded, just not in a way determined by the model).

Fix a basic language , and let be a countable set of new individual constants. For each let (so that ). We define the notion of truth by induction on -sentences, simultaneously for every . The quantifier case:

- An -sentence is true in the -structure if and only if the -sentence is true in every -expansion of (i.e., if is true in every -structure that differs from only in that it assigns a denotation to ).

So, when is an -sentence, the definition gives us a direct account of the truth or falsity of is a model having the same signature.

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The policy will go into effect on Nov. 1 for UCLA, UCI and UCSF, and a year later at all remaining campuses. More information can be found on the Open Access website and in the Frequently Asked Questions.

Needless to say, this is good all-around.

]]>Let . Intuitively, is the collection of all sets that only belong to well-founded sets. Clearly, is too big to be a set, but that is not the point. The point is that this can be shown in classical predicate logic using only existence of singletons:

We proceed by cases, and derive a contradiction in each.

- ; then only belongs to well-founded sets. Since exists and , the latter must be well-founded, so . But is equivalent to , so , which is impossible if (for then itself is in the intersection ).
- . Then there must be a such that: and . In particular , so there is a which belongs to both and . Now, since :
and since , we have contradicting the choice of .

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*Hint*: They do not speak French here.

And those are not the famous palm trees lining the *Champ de Mars*.

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